A position-measuring device having a cylindrical object and a stationary unit is known, for example, from European Patent Application EP 1 795 872 A1. FIG. 1 of EP 1 795 872 depicts a position-measuring device which includes, firstly, a cylindrical object which has a circumferential annular reflection measuring graduation and is rotatable about its longitudinal axis. Secondly, a stationary scanning unit is disposed opposite the rotatable object to optically scan the reflection measuring graduation. Suitable scanning units for such a device are shown, for example, in FIGS. 2B and 2C of EP 1 795 872 and each include a light source, a transmission grating, and a detector. The beams of light emitted by the light source pass through the transmission grating and then strike the reflection measuring graduation, from where they are reflected back toward the detector. The detector is capable of generating rotation-dependent position signals in response to relative motion between the reflection measuring graduation and the scanning unit.
The optical scanning principle used in such a position-measuring device is described in detail in the publication by R. M. Pettigrew entitled “Analysis of Grating Images and its Application to Displacement Metrology” in SPIE Vol. 136, 1st European Congress on Optics Applied to Metrology (1977), pp. 325-332. A brief description of this known scanning principle is given below:
A transmission grating is illuminated by a suitable light source such as, for example, an LED. Each illuminated line of the transmission grating emits a cylindrical wave toward the measuring standard, which is disposed at a distance u behind the transmission grating. Each of these cylindrical waves produces enlarged self-images of the measuring standard at distances v in the optical path. The periodicity dD of the self-images in a detection plane is given as follows:
                              d          D                =                              d            M                    ⁡                      (                          1              +                              v                u                                      )                                              (                  equation          ⁢                                          ⁢          1                )                dD=periodicity of the self-images of the measuring standard in a detection plane of the detector    dM=periodicity of the measuring standard    u=distance between the transmission grating and the measuring standard    v=distance between the measuring standard and the detection plane
By suitably selecting the periodicity dS of the transmission grating according to following equation 2, it is achieved that the self-images add up constructively incoherently on the detector.
                              d          S                =                              d            M                    ⁡                      (                          1              +                              u                v                                      )                                              (                  equation          ⁢                                          ⁢          2                )                dS=periodicity of the transmission grating    dM=periodicity of the measuring standard    u=distance between the transmission grating and the measuring standard    v=distance between the measuring standard and the detection plane
In the event that the measuring standard moves relative to the other components, the self-image of the measuring standard in the detection plane moves as well. If the detector is designed as a so-called structured photodetector, whose periodicity corresponds to that of the self-image of the measuring standard, then the detector can generate displacement-dependent position signals in the form of phase-shifted incremental signals. Moreover, for optimum contrast of the self-images of the measuring standard, it turns out to be advantageous if, in addition, the following condition (also known as “Talbot condition”) is met:
                              λ                      nd            M            2                          =                              1            u                    +                      1            v                                              (                  equation          ⁢                                          ⁢          3                )                dM=periodicity of the measuring standard    u=distance between the transmission grating and the measuring standard    v=distance between the measuring standard and the detection plane    n=1, 2, 3, . . . .
This scanning principle can be used particularly advantageously in reflected-light systems, where the measuring standard is designed as a reflection measuring graduation, and where the transmission grating and the detector are arranged in a common plane. In this case, the following holds:u=v  (equation 4)    u=distance between the transmission grating and the measuring standard    v=distance between the measuring standard and the detection plane
From equation 1, it follows that a change in the scanning distance (i.e., in a reflected-light system, distance u and v, respectively) will not cause a change in the periodicity dD of the self-images of the measuring standard, and therefore there will be no drop in the degree of modulation of the position signals.
These considerations apply strictly only to plane measuring standards. If the scanning principle described is to be used in position-measuring devices having curved measuring standards such as, for example, circumferential annular reflection measuring graduations, then it is not a priori guaranteed that the periodicity dD of the self-images of the measuring standard; i.e., the periodicity of the interference fringe pattern in the detection plane, does not change in response to a change in the scanning distance. The above-referenced European Patent Application EP 1 795 872 A1 does not contain any information on how to achieve in such position-measuring devices that the period of the fringe pattern in the detection plane is independent of the scanning distance.